Finite Group Databasebeta

← C2×C32×#806 #13162391

Subgroups — C2×C32×#806 #13162391

Order 1728.

1788 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#67C211yesmaximal, characteristic
2C2×C33×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×C32×#217C211yesmaximal, characteristic, complemented
4C32×#806C214yesmaximal, direct product factor
5C2×C32×#99—31nomaximal
6C2×C3×#806C314yesmaximal, direct product factor
7C24×C33C2211yescharacteristic
8C23×C32×D3C411yescharacteristic, complemented
9C22×C33×C4—21no—
10C2×C32×#67—21no—
11C2×C32×#67—21no—
12C2×C32×#67—21no—
13C2×C32×#67—21no—
14C2×C32×#67C412yescomplemented
15C2×C32×#67C2212yes—
16C22×C32×Dic3—21no—
17C22×C32×Dic3C411yescharacteristic, complemented
18C33×C22⋊C4—21no—
19C33×C22⋊C4C2212yescomplemented
20C32×#217—21no—
21C32×#217C2212yescomplemented
22C22×C32×D4—31no—
23C22×C3×#67C2×C314yes—
24C2×C32×C22⋊C4—24no—
25C2×C32×C22⋊C4—31no—
26C2×C32×C22⋊C4D311yescharacteristic, complemented
27C2×C32×C22⋊C4C2×C314yescomplemented
28C2×C3×#217C2×C314yescomplemented
29C32×#99—34no—
30C3×#806C2×C3116yesdirect product factor
31C23×C33—42no—
32C23×C33C2×C411yescharacteristic
33C23×C33C2×C411yescharacteristic
34C23×C33D411yescharacteristic
35C23×C33D411yescharacteristic
36C23×C33D411yescharacteristic
37C23×C33D411yescharacteristic
38C23×C33C2311yescharacteristic
39C22×C32×D3—21no—
40C22×C32×D3—21no—
41C22×C32×D3—42no—
42C22×C32×D3C2×C412yescomplemented
43C2×C33×C4—22no—
44C2×C33×C4—42no—
45C32×#67—22no—
46C32×#67—22no—
47C32×#67—22no—
48C32×#67—22no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3—22no—
52C2×C32×Dic3—42no—
53C2×C32×Dic3C2×C412yescomplemented
54C2×C3×#99—34no—
55C2×#806C3211yescharacteristic, direct product factor
56C24×C32—24no—
57C24×C32—31no—
58C24×C32C2×D311yescharacteristic
59C24×C32C22×C314yes—
60C23×C3×D3C3×C414yescomplemented
61C2×C32×D4—32no—
62C2×C32×D4—32no—
63C2×C32×D4—61no—
64C2×C32×D4—61no—
65C2×C32×D4—61no—
66C2×C32×D4—61no—
67C22×C32×C4—28no—
68C22×C32×C4—21no—
69C22×C32×C4—24no—
70C22×C32×C4—31no—
71C22×C32×C4—61no—
72C2×C3×#67—24no—
73C2×C3×#67—24no—
74C2×C3×#67—24no—
75C2×C3×#67—24no—
76C2×C3×#67C3×C418yescomplemented
77C2×C3×#67C22×C318yes—
78C22×C3×Dic3—24no—
79C22×C3×Dic3C3×C414yescomplemented
80C32×C22⋊C4—28no—
81C32×C22⋊C4—28no—
82C32×C22⋊C4—21no—
83C32×C22⋊C4—24no—
84C32×C22⋊C4—32no—
85C32×C22⋊C4—61no—
86C32×C22⋊C4C2×D312yescomplemented
87C32×C22⋊C4C22×C318yescomplemented
88C3×#217—24no—
89C3×#217C22×C318yescomplemented
90C22×C33—22no—
91C22×C33—22no—
92C22×C33—22no—
93C22×C33—41no—
94C22×C33—41no—
95C22×C33—41no—
96C22×C33—41no—
97C22×C33C22⋊C411yescharacteristic
98C22×C33C22⋊C411yescharacteristic
99C22×C33C22⋊C411yescharacteristic
100C22×C33C22⋊C411yescharacteristic
101C22×C33C22×C411yescharacteristic
102C22×C33C2×D411yescharacteristic
103C22×C33C2×D411yescharacteristic
104C2×C32×D3—22no—
105C2×C32×D3—42no—
106C2×C32×D3—42no—
107C2×C32×D3—42no—
108C33×C4—42no—
109C32×Dic3—22no—
110C32×Dic3—42no—
111C22×C3×D4—34no—
112C22×#67C2×C3211yescharacteristic
113C2×C3×C22⋊C4—24no—
114C2×C3×C22⋊C4—34no—
115C2×C3×C22⋊C4C3×D314yescomplemented
116C2×C3×C22⋊C4C2×C3211yescharacteristic, complemented
117C2×#217C2×C3211yescharacteristic, complemented
118C3×#99—316no—
119#806C2×C3214yesdirect product factor
120C23×C32—24no—
121C23×C32—24no—
122C23×C32—24no—
123C23×C32—24no—
124C23×C32—24no—
125C23×C32—24no—
126C23×C32—24no—
127C23×C32—32no—
128C23×C32—42no—
129C23×C32—48no—
130C23×C32—416no—
131C23×C32—61no—
132C23×C32—61no—
133C23×C32—122no—
134C23×C32C4×D311yescharacteristic
135C23×C32C4×D311yescharacteristic
136C23×C32D1211yescharacteristic
137C23×C32D1211yescharacteristic
138C23×C32#6711yescharacteristic
139C23×C32#6711yescharacteristic
140C23×C32C2×C3×C414yes—
141C23×C32C2×C3×C414yes—
142C23×C32C3×D414yes—
143C23×C32C3×D414yes—
144C23×C32C3×D414yes—
145C23×C32C3×D414yes—
146C23×C32C22×D311yescharacteristic
147C23×C32C23×C314yes—
148C22×C3×D3—24no—
149C22×C3×D3—24no—
150C22×C3×D3—48no—
151C22×C3×D3C2×C3×C418yescomplemented
152C32×D4—62no—
153C32×D4—62no—
154C32×D4—62no—
155C32×D4—62no—
156C2×C32×C4—216no—
157C2×C32×C4—22no—
158C2×C32×C4—28no—
159C2×C32×C4—32no—
160C2×C32×C4—416no—
161C2×C32×C4—42no—
162C2×C32×C4—48no—
163C2×C32×C4—61no—
164C2×C32×C4—61no—
165C2×C32×C4—62no—
166C2×C32×C4—122no—
167C3×#67—28no—
168C3×#67—28no—
169C3×#67—28no—
170C3×#67—28no—
171C2×C3×Dic3—24no—
172C2×C3×Dic3—24no—
173C2×C3×Dic3—28no—
174C2×C3×Dic3—48no—
175C2×C3×Dic3C2×C3×C418yescomplemented
176C2×#99—31noSylow
177C2×C33—21no—
178C2×C33—21no—
179C2×C33—21no—
180C2×C33—21no—
181C2×C33—21no—
182C2×C33—21no—
183C2×C33#9912yescomplemented
184C2×C33C2×C22⋊C411yescharacteristic
185C32×D3—42no—
186C24×C3—24no—
187C24×C3—34no—
188C24×C3C2×C3×D314yes—
189C24×C3C22×C3211yescharacteristic
190C23×D3C32×C411yescharacteristic, complemented
191C2×C3×D4—38no—
192C2×C3×D4—38no—
193C2×C3×D4—64no—
194C2×C3×D4—64no—
195C2×C3×D4—64no—
196C2×C3×D4—64no—
197C22×C3×C4—28no—
198C22×C3×C4—24no—
199C22×C3×C4—21no—
200C22×C3×C4—34no—
201C22×C3×C4—64no—
202C2×#67—21no—
203C2×#67—21no—
204C2×#67—21no—
205C2×#67—21no—
206C2×#67C32×C412yescomplemented
207C2×#67C22×C3212yes—
208C22×Dic3—21no—
209C22×Dic3C32×C411yescharacteristic, complemented
210C3×C22⋊C4—28no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—21no—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—38no—
215C3×C22⋊C4—64no—
216C3×C22⋊C4C2×C3×D318yescomplemented
217C3×C22⋊C4C22×C3212yescomplemented
218#217—21no—
219#217C22×C3212yescomplemented
220C22×C32—216no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—28no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—28no—
233C22×C32—28no—
234C22×C32—48no—
235C22×C32—48no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32—44no—
240C22×C32—44no—
241C22×C32—44no—
242C22×C32—41no—
243C22×C32—44no—
244C22×C32—48no—
245C22×C32—48no—
246C22×C32—48no—
247C22×C32—48no—
248C22×C32—62no—
249C22×C32—122no—
250C22×C32—122no—
251C22×C32—122no—
252C22×C32(C2×D3)⋊C411yescharacteristic
253C22×C32(C2×D3)⋊C411yescharacteristic
254C22×C32(C2×D3)⋊C411yescharacteristic
255C22×C32(C2×D3)⋊C411yescharacteristic
256C22×C32C3×C22⋊C414yes—
257C22×C32C3×C22⋊C414yes—
258C22×C32C3×C22⋊C414yes—
259C22×C32C3×C22⋊C414yes—
260C22×C32C2×C4×D311yescharacteristic
261C22×C32C2×D1211yescharacteristic
262C22×C32C2×#6711yescharacteristic
263C22×C32C22×C3×C414yes—
264C22×C32C2×C3×D414yes—
265C22×C32C2×C3×D414yes—
266C2×C3×D3—28no—
267C2×C3×D3—48no—
268C2×C3×D3—48no—
269C2×C3×D3—48no—
270C32×C4—416no—
271C32×C4—42no—
272C32×C4—48no—
273C32×C4—62no—
274C32×C4—122no—
275C3×Dic3—28no—
276C3×Dic3—48no—
277C22×D4—31no—
278C2×C22⋊C4—31no—
279C2×C22⋊C4C32×D311yescharacteristic, complemented
280#99—34no—
281C33C2×#9911yescharacteristic, complemented, Sylow
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—24no—
289C23×C3—38no—
290C23×C3—48no—
291C23×C3—42no—
292C23×C3—416no—
293C23×C3—64no—
294C23×C3—64no—
295C23×C3—128no—
296C23×C3C3×C4×D314yes—
297C23×C3C3×C4×D314yes—
298C23×C3C3×D1214yes—
299C23×C3C3×D1214yes—
300C23×C3C3×#6714yes—
301C23×C3C3×#6714yes—
302C23×C3C2×C32×C411yescharacteristic
303C23×C3C2×C32×C411yescharacteristic
304C23×C3C32×D411yescharacteristic
305C23×C3C32×D411yescharacteristic
306C23×C3C32×D411yescharacteristic
307C23×C3C32×D411yescharacteristic
308C23×C3C22×C3×D314yes—
309C23×C3C23×C3211yescharacteristic
310C22×D3—21no—
311C22×D3—21no—
312C22×D3—42no—
313C22×D3C2×C32×C412yescomplemented
314C3×D4—68no—
315C3×D4—68no—
316C3×D4—68no—
317C3×D4—68no—
318C2×C3×C4—216no—
319C2×C3×C4—28no—
320C2×C3×C4—22no—
321C2×C3×C4—38no—
322C2×C3×C4—416no—
323C2×C3×C4—48no—
324C2×C3×C4—42no—
325C2×C3×C4—64no—
326C2×C3×C4—64no—
327C2×C3×C4—68no—
328C2×C3×C4—128no—
329#67—22no—
330#67—22no—
331#67—22no—
332#67—22no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—42no—
337C2×Dic3C2×C32×C412yescomplemented
338C2×C32—28no—
339C2×C32—28no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—24no—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32—28no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32—44no—
355C2×C32—44no—
356C2×C32—44no—
357C2×C32—44no—
358C2×C32—122no—
359C2×C32#80612yesdirect product factor
360C2×C32C3×#9918yescomplemented
361C2×C32C2×(C2×D3)⋊C411yescharacteristic
362C2×C32C2×C3×C22⋊C414yes—
363C3×D3—48no—
364C24—31no—
365C24C2×C32×D311yescharacteristic
366C2×D4—32no—
367C2×D4—32no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C22×C4—21no—
373C22×C4—31no—
374C22×C4—61no—
375C22⋊C4—21no—
376C22⋊C4—32no—
377C22⋊C4—61no—
378C22⋊C4C2×C32×D312yescomplemented
379C22×C3—216no—
380C22×C3—28no—
381C22×C3—28no—
382C22×C3—28no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—24no—
389C22×C3—24no—
390C22×C3—24no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—48no—
394C22×C3—48no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—44no—
402C22×C3—41no—
403C22×C3—48no—
404C22×C3—48no—
405C22×C3—48no—
406C22×C3—48no—
407C22×C3—68no—
408C22×C3—128no—
409C22×C3—128no—
410C22×C3—128no—
411C22×C3C3×(C2×D3)⋊C414yes—
412C22×C3C3×(C2×D3)⋊C414yes—
413C22×C3C3×(C2×D3)⋊C414yes—
414C22×C3C3×(C2×D3)⋊C414yes—
415C22×C3C32×C22⋊C411yescharacteristic
416C22×C3C32×C22⋊C411yescharacteristic
417C22×C3C32×C22⋊C411yescharacteristic
418C22×C3C32×C22⋊C411yescharacteristic
419C22×C3C2×C3×C4×D314yes—
420C22×C3C2×C3×D1214yes—
421C22×C3C2×C3×#6714yes—
422C22×C3C22×C32×C411yescharacteristic
423C22×C3C2×C32×D411yescharacteristic
424C22×C3C2×C32×D411yescharacteristic
425C2×D3—22no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C3×C4—416no—
430C3×C4—48no—
431C3×C4—42no—
432C3×C4—68no—
433C3×C4—128no—
434Dic3—22no—
435Dic3—42no—
436C32—24no—
437C32C2×#80611yescharacteristic, direct product factor
438C32C2×C3×#9914yescomplemented
439C23—32no—
440C23—42no—
441C23—61no—
442C23—61no—
443C23—122no—
444C23C32×C4×D311yescharacteristic
445C23C32×C4×D311yescharacteristic
446C23C32×D1211yescharacteristic
447C23C32×D1211yescharacteristic
448C23C32×#6711yescharacteristic
449C23C32×#6711yescharacteristic
450C23C22×C32×D311yescharacteristic
451D4—62no—
452D4—62no—
453D4—62no—
454D4—62no—
455C2×C4—22no—
456C2×C4—32no—
457C2×C4—42no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—62no—
461C2×C4—122no—
462C2×C3—28no—
463C2×C3—28no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—24no—
469C2×C3—24no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—24no—
477C2×C3—28no—
478C2×C3—44no—
479C2×C3—44no—
480C2×C3—44no—
481C2×C3—44no—
482C2×C3—128no—
483C2×C3C3×#80618yesdirect product factor
484C2×C3C32×#9912yescomplemented
485C2×C3C2×C3×(C2×D3)⋊C414yes—
486C2×C3C2×C32×C22⋊C411yescharacteristic
487D3—42no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—62no—
496C22—122no—
497C22—122no—
498C22—122no—
499C22C32×(C2×D3)⋊C411yescharacteristic
500C22C32×(C2×D3)⋊C411yescharacteristic
501C22C32×(C2×D3)⋊C411yescharacteristic
502C22C32×(C2×D3)⋊C411yescharacteristic
503C22C2×C32×C4×D311yescharacteristic
504C22C2×C32×D1211yescharacteristic
505C22C2×C32×#6711yescharacteristic
506C4—42no—
507C4—62no—
508C4—122no—
509C3—24no—
510C3C2×C3×#80614yesdirect product factor
511C3C2×C32×#9911yescharacteristic, complemented
512C2—21no—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—21no—
517C2—21no—
518C2—122no—
519C2C32×#80612yesdirect product factor
520C2C2×C32×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22×C32(C2×D3)⋊C4
commutatorC22×C3C22×C32×C4
FrattiniC23C22×C32×D3
FittingC2×C33×C22⋊C4C2
socleC22×C33C22⋊C4
prime radicalC22×C32×#67C2
cosocleC23×C3C23×C32
hypercenterC2×C32×C22⋊C4D3
hypocenterC3C2×C32×#99
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C24×C32 ◁ C2×C32×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C33×C22⋊C4C2semidirect
3. C2×C32×#217C2semidirect
4. C32×#806C2direct (N × H)
6. C2×C3×#806C3direct (N × H)
8. C23×C32×D3C4semidirect
14. C2×C32×#67C4semidirect
17. C22×C32×Dic3C4semidirect
19. C33×C22⋊C4C22semidirect
21. C32×#217C22semidirect
26. C2×C32×C22⋊C4D3semidirect
27. C2×C32×C22⋊C4C2×C3semidirect
28. C2×C3×#217C2×C3semidirect
30. C3×#806C2×C3direct (N × H)
42. C22×C32×D3C2×C4semidirect
53. C2×C32×Dic3C2×C4semidirect
55. C2×#806C32direct (N × H)
60. C23×C3×D3C3×C4semidirect
76. C2×C3×#67C3×C4semidirect
79. C22×C3×Dic3C3×C4semidirect
86. C32×C22⋊C4C2×D3semidirect
87. C32×C22⋊C4C22×C3semidirect
89. C3×#217C22×C3semidirect
115. C2×C3×C22⋊C4C3×D3semidirect
116. C2×C3×C22⋊C4C2×C32semidirect
117. C2×#217C2×C32semidirect
119. #806C2×C32direct (N × H)
151. C22×C3×D3C2×C3×C4semidirect
175. C2×C3×Dic3C2×C3×C4semidirect
183. C2×C33#99semidirect
190. C23×D3C32×C4semidirect
206. C2×#67C32×C4semidirect
209. C22×Dic3C32×C4semidirect
216. C3×C22⋊C4C2×C3×D3semidirect
217. C3×C22⋊C4C22×C32semidirect
219. #217C22×C32semidirect
279. C2×C22⋊C4C32×D3semidirect
281. C33C2×#99semidirect
313. C22×D3C2×C32×C4semidirect
337. C2×Dic3C2×C32×C4semidirect
359. C2×C32#806direct (N × H)
360. C2×C32C3×#99semidirect
378. C22⋊C4C2×C32×D3semidirect
437. C32C2×#806direct (N × H)
438. C32C2×C3×#99semidirect
483. C2×C3C3×#806direct (N × H)
484. C2×C3C32×#99semidirect
510. C3C2×C3×#806direct (N × H)
511. C3C2×C32×#99semidirect
519. C2C32×#806direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C32×#672
C2×C33×C22⋊C42
C2×C32×#2172
C32×#8062
C2×C32×#993
C2×C3×#8063